论文标题
$ m_ {11} $和$ j_1 $扩展
Hasse norm principle for $M_{11}$ and $J_1$ extensions
论文作者
论文摘要
当galois组$ {\ rm gal}（\ rm gal}（l/k）$时，我们给出了$ k/k $的HASSE规范原则，我们给出了必要的条件通过确定$ h^1（k，{\ rm pic} \，\ edline {x}）= 0 $或$ \ mathbb {z}/2 \ mathbb {z} $ for norm norm norm norm norm norm one tori $ t = r^r^r^{（1）} _ {k/k/k}（k/k} $ \ OVERLINE {X} = X \ TIMES_K \ overline {k} $。结果为了解26美元的零星简单组的所有图片迈出了第一步。
We give a necessary and sufficient condition for the Hasse norm principle for field extensions $K/k$ when the Galois groups ${\rm Gal}(L/k)$ of the Galois closure $L/k$ of $K/k$ are isomorphic to the Mathieu group $M_{11}$ of degree $11$ of order $7920$ or the Janko group $J_1$ of order $175560$ by determining $H^1(k,{\rm Pic}\, \overline{X})=0$ or $\mathbb{Z}/2\mathbb{Z}$ for norm one tori $T=R^{(1)}_{K/k}(\mathbb{G}_m)$ with a smooth $k$-compactification $X$ and $\overline{X}=X\times_k\overline{k}$. The result gives a first step towards understanding the all pictures of the Hasse norm principle for the $26$ sporadic simple groups.
